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Greenhalgh, David and Rana, Sourav and Samanta, Sudip and Sardar,

Tridip and Bhattacharya, Sabyasachi and Chattopadhyay, Joydev (2015)

Awareness programs control infectious disease - Multiple delay induced

mathematical model. Applied Mathematics and Computation, 251. pp.

539-563. ISSN 0096-3003 , http://dx.doi.org/10.1016/j.amc.2014.11.091

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Awareness programs control infectious disease - Multiple delay induced1

mathematical model.2

David Greenhalgh1, Sourav Rana2, Sudip Samanta3, Tridip Sardar3, Sabyasachi Bhattacharya3, Joydev3

Chattopadhyay3,44

Abstract5

We propose and analyze a mathematical model to study the impact of awareness programs on an infec-6

tious disease outbreak. These programs induce behavioral changes in the population, which divide the7

susceptible class into two subclasses, aware susceptible and unaware susceptible. The system can have a8

disease-free equilibrium and an endemic equilibrium. The expression of the basic reproduction number9

and the conditions for the stability of the equilibria are derived. We further improve and study the model10

by introducing two time-delay factors, one for the time lag in memory fading of aware people and one for11

the delay between cases of disease occurring and mounting awareness programs. The delayed system has12

positive bounded solutions. We study various cases for the time delays and show that in general the sys-13

tem develops limit cycle oscillation through a Hopf bifurcation for increasing time delays. We show that14

under certain conditions on the parameters, the system is permanent. To verify our analytical findings,15

the numerical simulations on the model, using realistic parameters for Pneumococcus are performed.16

Keywords: Epidemic model, Awareness programs, Time delay, Stability analysis, Hopf bifurcation,17

Numerical simulation.18

Mathematics Subject Classification: 34D20, 92B05, 92D20, 92D39.

1Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, GlasgowG1 1XH, UK, Email: david.greenhalgh@strath.ac.uk. Tel.: +44-141-548-3653, Fax: +44-141-548-3345

2Department of Statistics, Visva-Bharati University, Santiniketan, West Bengal, Pin 731235, India3Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India4Corresponding author. E-mail: joydev@isical.ac.in, Fax: +91-33-25773049, Tel: +91-33-25753231.

1. Introduction19

In developing countries more than 11 million people died each year due to infectious disease includ-20

ing premature deaths and deaths of young children. Pneumonia, Tuberculosis (TB), Diarrheal diseases21

(Cholera), Malaria, Measles and more recently HIV/AIDS are the major deadly infectious diseases [1].22

The major cause of global childhood mortality is Pneumonia which is caused by a number of infectious23

agents, including viruses, bacteria and fungi. Approximately 1.4 million children die every year because24

of Pneumonia [2]. Diarrheal diseases (for example Cholera, Bacillary Dysentery, Typhoid, Giardia and25

Rotavirus) are the second leading cause of death taking the lives of about 1.5 million children under five26

every year [3]. In 2010, 8.8 million people were infected with, and 1.4 million died from, TB [4]. Malaria is27

a life-threatening vector-borne disease caused by the bites of infected mosquitoes. In 2010, Malaria caused28

an estimated 655,000 deaths, mostly among African children (with an uncertainty range of 537,000 to29

907,000) [5]. In 2010, 139,300 people died worldwide due to Measles [6]. Recently, HIV/AIDS has become30

the major concern in a global pandemic. More than 25 million people died of HIV/AIDS in the last three31

decades. There were approximately 34.2 million people infected by HIV up to the end of 2011 [7]. Another32

infectious disease is Influenza which causes serious public health and economic problems. Globally, these33

annual epidemics result in about three to five million cases of severe illness, and about 250,000 to 500,00034

deaths [8]. Other major deadly infectious diseases in humans include Dengue, Yellow Fever, Hepatitis B,35

Avian Influenza (Bird Flu) and Chagas Disease.36

The above description clearly indicates the severity of infectious disease. These diseases are a major37

threat to developing and underdeveloped countries. Some diseases can be prevented through vaccina-38

tions. However this is costly and sometimes the effect is only temporary. On the other hand sometimes39

disease appropriate awareness in a population can control an infection most effectively. In developing40

and underdeveloped countries, the mass media plays an important role in changing behavior related to41

public health. The government and other health organizations should immediately make people aware42

about the disease and relevant precautions through the media. The media not only make the population43

acquainted with the disease but also suggest the necessary preventive practices such as social distancing,44

wearing protective masks or vaccination. In general the people who are aware adopt these practices so45

that their chances of becoming infected are minimized. Depending on the behavior associated with a46

given infectious disease, improved levels of awareness may increase the use of mosquito coils, mosquito47

nets [9], or face masks [10, 11], practice of better hygiene [12, 13], application of preventive medicine48

or vaccination [14], voluntary quarantine [15], avoidance of places containing large numbers of people49

2

[12], practice of safe sex [16], or other appropriate measures. A comprehensive review of the existing50

mathematical literature related to the effect of media awareness programs on disease outbreaks is given in51

Table 1. However, behavioral responses can change the transmission patterns and reduce the prevalence52

of disease. So there is a need of epidemiological models that explicitly include the effect of awareness53

programs and behavioral responses. It is to be noted that in general the effect of awareness can strongly54

depend on local interactions. The individuals in the local spatial or geographical neighbourhood of an55

outbreak may have a much stronger incentive to adopt preventive practices and this local adoption of56

suitable preventive practices may cause a local outbreak to die out without the whole population having57

to adopt them. It would be possible to model this using some sort of spatial model. However in this58

paper we shall not pursue this line instead we shall study a mean field model and assume that the impact59

of the awareness program is uniform across the whole population. This is common in the study of disease60

awareness programs [17, 18, 19, 20] where sometimes we wish to use a relatively simple model to study61

the effect of awareness programs applied to the whole population to reduce the disease levels in the entire62

population rather than stop a local outbreak.63

A comprehensive review on the impact of media awareness programmes is presented in Section 2. In64

Section 3 the model without time delays is formulated and analyzed to observe the local stability of the65

system around the feasible equilibria. The model with multiple time delays is proposed and analyzed in66

Section 4. The conditions under which the system enters Hopf bifurcation and conditions for permanence67

of the system are also worked out. In Section 5, numerical simulations are carried out to verify our68

analytical findings and the paper ends with a brief conclusion.69

2. Review of media awareness program in infectious disease outbreak70

In this section we review the literature on the effect of media awareness programs on infectious71

disease outbreaks. These studies are essentially of two different types. In the first type mathemati-72

cal models are used to investigate the impact of media coverage on the spread and control of infec-73

tious disease. The mathematical models are either compartmental models such as susceptible-infected-74

susceptible (SIS), susceptible-infected-recovered (SIR), susceptible-exposed-infected (SEI), susceptible-75

infected-recovered-susceptible (SIRS), exposed-infected-hospitalized (EIH), susceptible-exposed-infected-76

hospitalized-recovered (SEIHR) and similar models, or economic or game-theoretic models. In the second77

type of study statistical analysis is used to identify the association between media awareness and disease78

related cases. A comprehensive summary of such studies is given in Table 1.79

3

Table 1: Review on the impact of media awareness programs

on infectious disease.

Year References Summary of study

2007 [21] Cui et al. developed and analyzed an SEI model to include media influ-

ence on the spreading of a communicable disease in a given area. They

concluded that if the basic reproduction number is greater than one and

the media effect is high, the model shows several endemic equilibria,

which causes a threat to control the disease outbreak.

[17] Liu et al. developed an EIH compartmental model to investigate the role

of the media and its psychological impact on multiple disease outbreaks.

Their model analysis reveals that this impact leads to differences in the

transmission pattern.

[22] Using the data from the Bangladesh Demography and Health Survey

(1999-2000), Rahman and Rahman identified that media and education

could play a major role in controlling HIV/AIDS.

[23] Tai and Sun investigated media dependency amongst Chinese individu-

als during the SARS epidemic of 2003. Their study was mainly focused

into the situation where the information was highly monitored and not

easily available from the mainstream media. In those circumstances,

short message service (SMS) and the Internet are the possible substi-

tute resources of information.

2008 [24] Cui et al. formulated and analyzed an SIS infection model to investi-

gate the role of media coverage during an infectious disease outbreak

in a given population. They concluded that increasing media coverage

causes a lower infection rate, although it may not absolutely remove the

infection.

4

Table 1 – continued from previous page

Year References Summary of study

[25] Joshi et al. investigated the effect of an information and education

campaign on the HIV epidemic in Uganda. They compare their model

with three types of susceptibles to a standard SIR model.

[26] Li et al. developed and analyzed an SIS epidemic model, including me-

dia coverage in which the susceptible population is subjected to impul-

sive vaccination. They showed that the disease-free solution is globally

asymptotically stable.

[27] Liu and Cui developed a compartmental model to study the role of

the media in an infectious disease outbreak. They assume a standard

epidemiological model but with a reduced transmission term due to the

media campaign.

[28] Young et al. showed that a high level of media coverage plays a crucial

role in making the public aware of many diseases and influencing their

perception of risk. Participants in their study often considered diseases

that appeared in the media more serious, even when this was not the

actual case.

2009 [29] Chen formulated an economic game-theoretic model of epidemics incor-

porating self-protection of susceptible populations. He suggests that an

individual makes his or her behavioral changes through the information

about the disease and expanding the supply of information may decrease

the likelihood of eradication.

5

Table 1 – continued from previous page

Year References Summary of study

[30] Funk et al. develop and study a mathematical model where the host

population is less susceptible due to the spread of awareness. They reveal

that change in behavioral response can reduce the size of an outbreak

though the epidemic threshold will be unaffected.

[31] Li and Cui propose an SIS epidemic model in the presence of media

coverage and analyze the model under two distinct types of vaccina-

tion strategies namely constant vaccination and pulse vaccination. They

compare these two different types of vaccination policies.

2010 [32] Kiss et al. formulated a mathematical model where the total popula-

tions are aware of the disease threat but only a certain proportion of

them is responsive. They showed that the infection can be removed

when the spreading of information is fast enough, otherwise information

transmission can play a major role in controlling the disease.

[33] Mummert and Weiss proposed a modified SIR model incorporating the

impact of media coverage. They conclude that the severity of the disease

outbreak can be lower if the media and the public health agencies work

together.

[34] Yoo et al. showed using a statistical analysis that there is a connection

between Influenza vaccination 1999-2001 and media reporting, specifi-

cally headlines on flu-related issues. They studied three media sources:

a wire service news agency, a newspaper and four television channels.

6

Table 1 – continued from previous page

Year References Summary of study

2011 [18] Misra et al. developed and analyzed a nonlinear SIS mathematical model

in the presence of a media awareness program. They suggest that an

awareness program can control the diffusion of the disease but immigra-

tion of susceptibles causes the disease to be endemic.

[35] Misra et al. proposed and analyzed a delay induced mathematical model

in the presence of an awareness program. They concluded that the

awareness program plays a crucial role in controlling the spread of dis-

ease, but it cannot remove the infection completely.

[36] Sun et al. used the SIS model in a two patch setting with media coverage

present in each patch. They analyze their model both analytically and

numerically. They find that both epidemic burden and duration of the

disease spread are significantly lowered by the media coverage.

[19] Tchuenche et al. developed a Susceptible-Infected-Vaccinated-

Recovered (SIVR) epidemic model to study the effect of media broad-

casting on the spread and control over an Influenza outbreak. Using

optimal control theory they obtained the effect of costs due to media

coverage.

2012 [37] Olowukure et al. investigated if there is any connection between volume

of newspaper reports and laboratory testing for Influenza A (H1N1)

pdm09, (the swine flu Influenza A (H1N1) pandemic of 2009) in one

English health region during the early phase of the pandemic. They in-

ferred that there exists a temporal association between volume of media

reporting and number of laboratory tests.

7

Table 1 – continued from previous page

Year References Summary of study

[38] Tchuenche and Bauch formulated an SIHR model incorporating a signal

function which captures the effect of media coverage. They suggest that

the disease cannot be eliminated through media coverage, but it can

control the spread of the infection.

2013 [39] Funk and Jansen studied how the interplay between the network of an

awareness program and the network of infection determines the dynam-

ics of the disease outbreak.

[40] Liu investigated an SIRS epidemic model with media coverage and ran-

dom perturbation. The disease transmission term was reduced by media

coverage as in Liu and Cui [27], Tchuenche et al. [19] and Sun et al.

[36] and stochastic white noise perturbation was added. The result-

ing stochastic differential equation model was studied analytically and

numerically.

[20] Samanta et al. studied an SIS epidemic model for the effect of media

awareness programs on epidemic outbreaks. They concluded that al-

though media awareness programs can have a substantial effect on con-

trolling disease prevalence, above a threshold value of their execution

rate, the system shows limit cycle oscillations.

[41] Wang et al. studied an SIS network model incorporating the impact

of media coverage on disease transmission and suggested effective con-

trol strategies to prevent disease through media coverage and education.

They find the basic reproduction number, equilibrium and global stabil-

ity results for their model and explore the results by simulation.

8

Table 1 – continued from previous page

Year References Summary of study

2014 [42] Kaur et al. proposed and analyzed an SIRS epidemic model incorpo-

rating the effects of an awareness program driven by the media. Their

model is based on that of Misra et al. [18] with some significant differ-

ences in modeling the awareness programs. They conduct an equilibrium

and stability analysis and use simulation to verify their results.

[43] Samanta and Chattopadhyay proposed and analyzed a slow-fast epi-

demic model in the presence of the awareness program, where a suscep-

tible individual switches between aware and unaware states very fast,

whereas the disease transmission and other biological processes are com-

paratively slow.

[44] Sharma and Misra investigated an SIR model of hepatitis B with varying

population size, which couples vaccination and awareness created by the

media within a single framework.

[45] Wang and Xiao studied an SIR Filippov epidemic model with media

coverage by incorporating a piecewise continuous transmission rate to

describe that the media coverage exhibits its effects once the number of

infected individuals exceeds a certain critical level. The disease transmis-

sion coefficient is reduced by an exponential term as a result of a media

campaign. They find that a given level of infecteds can be reached if the

threshold policy and other parameters are chosen correctly.

9

Table 1 – continued from previous page

Year References Summary of study

[46] Zhao et al. proposed and analyzed an SIRS epidemic model incorporat-

ing media coverage with time delay. They showed that the time delay in

media coverage cannot affect the stability of the disease-free equilibrium

when the basic reproduction number is less than unity. However, the

time delay affects the stability of the endemic equilibrium and produces

limit cycle oscillations while the basic reproduction number is greater

than unity.

2015 [47] Sahu and Dhar studied the complex dynamics of an SEQIHRS epidemic

model incorporating media coverage, quarantine and isolation studies in

a community with pre-existing immunity. Media coverage does not alter

the effective reproduction number but lowers the number of infecteds

at the endemic steady state, also lowering the maximum number of

infected individuals. The results of isolation and quarantine depend on

the amount of transmission from isolated individuals. Higher amounts of

pre-existing immunity amongst the population cause the peak infection

level to happen earlier and decrease it.

80

The above descriptions clearly indicate that awareness programmes play a crucial role in controlling81

the disease during an epidemic outbreak. In the next section we formulate a mathematical model to82

capture the impact of media awareness programs in an infectious disease outbreak. The model that we83

shall consider is a deterministic differential equation mean field SIS epidemic model for the spread of an84

infection in the presence of awareness programs. We model the awareness programs explicitly unlike the85

models of Cui et al. [24], Li, Ma and Cui [26] and Liu and Cui [27] who model the effect of awareness86

through a reduction in the disease transmission term. Our work builds on the work of Misra et al. [18, 35]87

although we allow aware people to become infected and some recovered individuals to become aware. It88

also builds on Samanta et al. [20] After analysing the basic model we introduce and analyse two types of89

time delays and then perform simulations based on real parameter values for Pneumococcus to verify our90

10

theoretical results.91

3. Model with awareness program92

3.1. Model Formulation93

To formulate the mathematical model we suppose that the whole population is divided into three94

separate classes, the susceptible aware class, the susceptible unaware class and the infected class. We95

assume that both susceptible classes can be infected by contact with infectives but the aware class has96

less chance to be infected compared to the unaware class and the infection rate among aware populations97

is dependent on the awareness programs. The unaware susceptible population becomes aware through98

the interaction with the awareness programs [18, 35] which is considered to be a saturating function [27]99

(Holling type-II) of the awareness programs and a proportion of infected individuals recover from the100

infection through treatment. After recovery, a fraction of recovered people will join the aware susceptible101

class and the remaining fraction will remain unaware susceptible. The model does not necessarily assume102

that the transmission routes of the disease and the information are the same, indeed these may well be103

different.104

We consider that in the region under consideration, the total population is N(t) at time t and the rate105

of immigration of susceptibles is A, where immigrants are assumed to be unaware. The total population106

is divided into three classes: the susceptible unaware population X−(t), the infective population Y (t)107

and the susceptible aware population X+(t). Also, let M(t) be the number of campaigns due to the108

awareness programs driven by the media in that region at time t. µ denotes the implementation rate of109

awareness programs which is proportional to the number of infective individuals in the population. We110

assume that unaware susceptible individuals become aware under the influence of the awareness program111

at the rate λ and the interaction between the unaware susceptible population and the awareness program112

follows the Holling type-II functional form with half-saturating constant k. It is assumed that the disease113

spreads only due to direct contact between susceptibles and infectives. Let β be the contact rate of114

unaware susceptible individuals with infective individuals and it is assumed that the disease transmission115

follows the mass action law (βX−(t)Y (t)). However, our basic assumption is that the interaction between116

aware susceptibles and infecteds depends on the number of campaigns due to the awareness programs.117

Large numbers of campaigns causes less interaction between susceptible aware and infected populations,118

a mathematical form of this assumption can be written as βX+(t)Y (t)1+β1M(t) , where β1 is the efficacy of the119

awareness programs - a monotonic decreasing function of the number of campaigns M(t). It is also a120

monotonic decreasing function of β1. We assume that aware susceptible individuals transfer to unaware121

11

susceptible individuals due to fading of memory or social factors at a per capita rate λ0. It is also122

assumed that a proportion of infected individuals recover through treatment. After recovery, a fraction p123

of recovered people will become aware and join the aware susceptible class whereas the remaining fraction124

(1 − p) will remain unaware susceptible.125

Keeping the above facts in mind, the dynamics of the model is governed by the following systems of126

nonlinear ordinary differential equations :127

dX−

dt= A− βX−(t)Y (t) − λX−(t)

M(t)

k +M(t)− dX−(t) + λ0X+(t) + (1 − p)γY (t),

dX+

dt= λX−(t)

M(t)

k +M(t)+ pγY (t) − dX+(t) − λ0X+(t) −

β

1 + β1M(t)X+(t)Y (t),

dY

dt= βX−(t)Y (t) +

β

1 + β1M(t)X+(t)Y (t) − γY (t) − αY (t) − dY (t),

dM

dt= µY (t) − µ0M(t),

(3.1)

where X−(0) > 0, X+ ≥ 0, Y ≥ 0, M ≥ 0.128

Here the constants γ, α, d represent the recovery rate, disease induced death and natural death rate129

respectively. The constant µ0 denotes the depletion rate of awareness programs due to ineffectiveness,130

social problems in the population, and similar factors. Note that p is a fraction and its value lies between131

0 and 1.132

Using the fact N = X− +X+ + Y , the system (3.1) reduces to the following system:133

dY

dt= β(N(t) −X+(t) − Y (t))Y (t) +

β

1 + β1M(t)X+(t)Y (t) − (γ + α+ d)Y (t),

dX+

dt= λ(N(t) −X+(t) − Y (t))

M(t)

k +M(t)+ pγY (t) − dX+(t) − λ0X+(t)

−β

1 + β1M(t)X+(t)Y (t),

dN

dt= A− dN(t) − αY (t),

dM

dt= µY (t) − µ0M(t).

(3.2)

12

For the analysis of model (3.2), we need the region of attraction [48] which is given by the set:

Ω =

(Y,X+, N,M) ∈ ℜ+4 : 0 ≤ X+ + Y ≤ N ≤

A

d, 0 ≤M ≤

µA

µ0d

and attracts all solutions initiating in the interior of the positive orthant, with N(0) > X+(0) + Y (0).134

3.2. Equilibrium analysis135

The above model (3.2) has two non-negative equilibria.136

(i) The disease free equilibrium (DFE) E0(0, 0, A/d, 0).137

(ii) The endemic equilibrium E∗(Y ∗,X∗+, N

∗,M∗).138

Here139

X∗+ =

µ0

ββ1µY ∗

[

β(A

d−αY ∗

d− Y ∗

)

−(

γ + α+ d)

][

1 +β1µY

∗

µ0

]

,140

N∗ =A− αY ∗

d,141

M∗ =µY ∗

µ0,142

and Y ∗ satisfies the equation143

H1Y∗3 +H2Y

∗2 +H3Y∗ +H4 = 0, (3.3)

with144

H1 = ββ1µ2

µ20

[(

d+ λ0

)(

αd

+ 1)

+ pγ]

+ β2µµ0

(

αd

+ 1)

,

H2 = β(αd

+ 1)(λµµ0

+ βk) − (βAd

− γ − α− d)(λβ1µ2

µ20

+ βµµ0

) + λββ1µ2A

µ20d

+pγββ1µkµ0

− β1µ2

µ20

(d+ λ0)(βAd

− γ − α− d) + βµµ0

(d+ λ0)(1 + β1k)(αd

+ 1),

H3 = −(

βAd

− γ − α− d)(

λµµ0

+ βk)

+kβ(

d+ λ0

)(

αd

+ 1)

− µµ0

(

d+ λ0

)(

1 + β1k)(

βAd

− γ − α− d)

,

H4 = −k(

βAd

− γ − α− d)(

d+ λ0

)

.

(3.4)

An endemic equilibrium exists if145

βA

d− (γ + α+ d) > 0. (3.5)

Let us define R0 =βA

d(γ + α+ d), which is the basic reproduction number for system (3.2).146

H1 is always positive and H4 is always negative if R0 > 1. Hence the equation (3.3) has at least147

one positive root. Therefore the sufficient conditions for the existence of the interior equilibrium point of148

system (3.2) are as follows:149

R0 > 1 and Y ∗ < min

d(γ+α+d)(R0−1)β(α+d) , A

α

.

13

However, H1, H2, H3 and H4 are always positive if R0 < 1. Hence the system (3.2) does not have any150

positive interior equilibrium (E∗) for R0 < 1.151

Remark 1: ∂Y ∗

∂µ< 0 if

H1µY ∗2+H2µY ∗+H3µ

3H1Y ∗2+2Y ∗H2+H3> 0 and ∂Y ∗

∂β1< 0 if

H1β1Y ∗2+H2β1

Y ∗+H3β1

3H1Y ∗2+2Y ∗H2+H3> 0,152

which indicates that the equilibrium number of infective individuals decreases with an increase in the153

value of the the implementation rate of awareness programs and the efficacy of the awareness programs.154

Here Hi•, (i = 1, 2, 3) denotes the partial differentiation of Hi with respect to the parameter ’•’.155

Remark 2: We can find the basic reproduction number of the system (3.1) in the absence of awareness156

program. Therefore the system (3.1) becomes157

dSdt

= A− βSY − dS + γY,

dYdt

= βSY − γY − αY − dY,(3.6)

where S and Y are the number of susceptible and infected individuals and the other parameters are the158

same as defined in system (3.1).159

The above model (3.6) has two non-negative equilibria:160

(i) The disease free equilibrium (DFE) E0(0, A/d),161

(ii) The endemic equilibrium E∗(S∗, Y ∗),162

where S∗ = γ+α+dβ

, Y ∗ = βA−d(γ+α+d)β(α+d) the basic reproduction number for the system (3.6) is R01 =163

βAd(γ+α+d) , which is the same as R0. So the awareness program cannot eradicate the infection whenever164

R0 > 1, but it can reduce the equilibrium number of infected individuals (see Figure 2).165

3.3. Local stability behavior166

The roots of the characteristic equation corresponding to E0(0, 0, A/d, 0) are βAd

− γ − α − d, −d,167

−(d+ λ0), −µ0.168

The DFE E0 is locally asymptotically stable (LAS) if βAd

− γ − α− d < 0, i.e. R0 < 1.169

The variational matrix at an endemic equilibrium E∗(Y ∗,X∗+, N

∗,M∗) is170

J =

−Π1 − ξ Π2 Π3 −Π4

Π5 −Π6 − ξ Π7 Π8

−Π9 0 −Π10 − ξ 0

Π11 0 0 −Π12 − ξ

.

Here Π1 = βY ∗, Π2 = −βY ∗ + βY ∗

1+β1M∗ , Π3 = βY ∗, Π4 =ββ1X∗

+Y ∗

(1+β1M∗)2, Π5 = − λM∗

k+M∗ + pγ −βX∗

+

1+β1M∗ ,171

Π6 = λM∗

k+M∗ + d+ λ0 + βY ∗

1+β1M∗ , Π7 = λM∗

k+M∗ , Π8 =λ(N∗−X∗

+−Y )k

(k+M∗)2 +ββ1X∗

+Y ∗

(1+β1M∗)2 , Π9 = α, Π10 = d, Π11 = µ,172

Π12 = µ0.173

14

The characteristic equation of the system (3.2) around the interior equilibrium (E∗) is174

ξ4 + σ1ξ3 + σ2ξ

2 + σ3ξ + σ4 = 0. (3.7)

Therefore, E∗ is LAS if and only if175

σ1 > 0, σ2 > 0, σ3 > 0, σ4 > 0, σ1σ2 > σ3 and σ1σ2σ3 > σ23 + σ2

1σ4. (3.8)

Here,176

σ1 = Π1 + Π6 + Π10 + Π12,177

σ2 = Π1Π10 + Π1Π12 + Π10Π12 + Π3Π9 + Π4Π11 + Π6Π10 + Π6Π12 + Π1Π6 − Π2Π5,178

σ3 = −Π2Π5Π10 − Π2Π5Π12 + Π2Π8Π11 + Π1Π10Π12 + Π3Π9Π12 + Π4Π10Π11 + Π6Π10Π12179

+Π1Π6Π10 + Π1Π6Π12 + Π3Π6Π9 + Π4Π6Π11 + Π2Π7Π9,180

σ4 = −Π2Π5Π10Π12 + Π2Π7Π9Π12 − Π2Π8Π10Π11 + Π1Π6Π10Π12 + Π3Π6Π9Π12 + Π4Π6Π10Π11.181

4. Model with delay182

4.1. Model Formulation183

In the previous section we assumed that aware susceptible individuals transfer to unaware susceptible184

individuals due to fading of memory or certain social factors. However, it is reasonable to consider a time185

lag in memory fading of aware people. Here we assume that the aware susceptible individual will become186

unaware susceptible at time t due to forgetting the impact of disease at time t− τ1 (for some τ1 > 0).187

We need to consider the probability that an aware susceptible individual remains in the aware suscep-188

tible class throughout the interval [t− τ1, t] which we denote by P (t, τ1). An aware susceptible individual189

leaves the aware susceptible class at time ξ through death at rate d, surviving the time interval [ξ− τ1, ξ]190

and becoming unaware at rate λ0P (ξ, τ1) or becoming infected at rate βY (ξ)1+β1M(ξ) . Hence191

P (t, τ1) = e−

R tt−τ1

[

d+λ0P (ξ,τ1)+βY (ξ)

1+β1M(ξ)dξ

]

, for t ≥ t1. (4.1)

Usually, the number of infective cases known to the policy makers are cases that occurred some time192

previously and thus the intensity of the awareness program depends on this data. So it is more plausible193

to consider a time delay in execution of awareness programs. We suppose that at time t the intensity of194

the awareness programs being executed will be in accordance with the number of infected cases reported195

at time t− τ2 (for some τ2 > 0).196

Incorporating these two delays and the survival probability into the system of equations (3.1) and197

writing P (t) ≡ P (t, τ1) as τ1 is fixed we obtain the system of delay differential equations:198

15

dX−

dt= A− βX−(t)Y (t) − λX−(t)

M(t)

k +M(t)− dX−(t) + λ0X+(t− τ1)P (t) + (1 − p)γY (t),

dX+

dt= λX−(t)

M(t)

k +M(t)+ pγY (t) − dX+(t) − λ0X+(t− τ1)P (t) −

β

1 + β1M(t)X+(t)Y (t),

dY

dt= βX−(t)Y (t) +

β

1 + β1M(t)X+(t)Y (t) − γY (t) − αY (t) − dY (t),

dM

dt= µY (t− τ2) − µ0M(t),

dP

dt=

[

−λ0P (t) + λ0P (t− τ1) −βY (t)

1 + β1M(t)+

βY (t− τ1)

1 + β1M(t− τ1)

]

P (t).

(4.2)

We denote by C the Banach space of continuous functions φ : [−τ, 0] → R5 with norm

‖φ‖ = sup−τ≤θ≤0

|φ1(θ)|, |φ2(θ)|, |φ3(θ)|, |φ4(θ)|, |φ5(θ)|

where τ = maxτ1, τ2 and φ = (φ1, φ2, φ3, φ4, φ5). As usual, the initial conditions of (4.2) are given as199

X−(θ) = φ1(θ), X+(θ) = φ2(θ), Y (θ) = φ3(θ), M(θ) = φ4(θ), P (θ) = φ5(θ), θ ∈ [−τ, 0], (4.3)

where the initial function φ = (φ1, φ2, φ3, φ4, φ5) belongs to the Banach space C = C([−τ, 0],R5) of200

continuous functions mapping the interval [−τ, 0] into R5. For biological reasons, the initial functions are201

assumed as202

φi(θ) ≥ 0, i = 1, 2, 3, 4 and 1 ≥ φ5(θ) ≥ 0, θ ∈ [−τ, 0]. (4.4)

We also need the consistency condition203

P (0) = e−

R 0−τ

[

d+λ0P (ξ,τ1)+βY (ξ)

1+M(ξ)

]

dξ.

By the fundamental theory of functional differential equations [49], we know that there is a unique204

solution (X−(t),X+(t), Y (t),M(t), P (t)) to system (4.2) with initial conditions (4.3).205

4.2. Preliminaries206

In this section, we will present some preliminaries, such as positive invariance, boundedness of solu-207

tions, existence of equilibria and the characteristic equation.208

16

4.2.1. Positive invariance209

Theorem 4.1. All the solutions of (4.2) with initial conditions (4.3) are positive.210

Proof : The model (4.2) can be written in the following form:211

X = col(X−(t),X+(t), Y (t),M(t), P (t)) ∈ R5+, (φ1(θ), φ2(θ), φ3(θ), φ4(θ), φ5(θ)) ∈ C+ = ([−τ, 0],R5

+),212

φ1(0), φ2(0), φ3(0), φ4(0) ≥ 0, φ5(0) ≥ 0,213

F (X) =

F1(X)

F2(X)

F3(X)

F4(X)

F5(X)

=

A− βX−(t)Y (t) − λX−(t) M(t)k+M(t) − dX−(t) + λ0X+(t− τ1)P (t) + (1 − p)γY (t)

λX−(t) M(t)k+M(t) + pγY (t) − dX+(t) − λ0X+(t− τ1)P (t) − β

1+β1M(t)X+(t)Y (t)

βX−(t)Y (t) + β1+β1M(t)X+(t)Y (t) − γY (t) − αY (t) − dY (t)

µY (t− τ2) − µ0M(t)[

−λ0P (t) + λ0P (t− τ1) −βY (t)

1+β1M(t) + βY (t−τ1)1+β1M(t−τ1)

]

P (t)

.

Then the model system (4.2) becomes214

X = F (X) (4.5)

with X(θ) = (φ1(θ), φ2(θ), φ3(θ), φ4(θ), φ5(θ)) ∈ C+ and φ1(0), φ2(0), φ3(0), φ4(0), φ5(0) > 0. It is easy to

check in system (4.5) that whenever choosing X(θ) ∈ R+ such that X− = 0,X+ = 0, Y = 0,M = 0 or

P = 0 then

Fi(X)|xi=0,X∈R5+≥ 0, for i = 1, 2, 3, 4, 5,

with x1(t) = X−(t), x2(t) = X+(t), x3(t) = Y (t), x4(t) = M(t), x5(t) = P (t). Using the lemma of [50]215

we claim that any solution of (4.5) with X(θ) ∈ C+, say X(t) = X(t,X(θ)), is such that X(t) ∈ R5+ for216

all t ≥ 0. From (4.1) we can see that P (t) ≤ 1 for all t as well.217

Next, we will prove the boundedness of solutions. Using the fact N = X− +X+ +Y , the system (4.2)218

reduces to the following system:219

17

dY

dt= β(N(t) −X+(t) − Y (t))Y (t) +

β

1 + β1M(t)X+(t)Y (t) − (γ + α+ d)Y (t),

dX+

dt= λ(N(t) −X+(t) − Y (t))

M(t)

k +M(t)+ pγY (t) − dX+(t) − λ0X+(t− τ1)P (t)

−β

1 + β1M(t)X+(t)Y (t),

dN

dt= A− dN(t) − αY (t),

dM

dt= µY (t− τ2) − µ0M(t),

dP

dt=

[

−λ0P (t) + λ0P (t− τ1) −βY (t)

1 + β1M(t)+

βY (t− τ1)

1 + β1M(t− τ1)

]

P (t).

(4.6)

4.2.2. Boundedness220

Theorem 3.2. All the solutions of (4.6) with initial conditions (4.3) are ultimately bounded.221

Proof : Let, (Y (t),X+(t), N(t),M(t), P (t)) be any solution of system (4.6) with initial conditions (4.3).222

Applying the theorem of differential inequality [51] on the third equation of the system (4.6), we have223

N(t) ≤ e−dt(

N(0) − Ad

)

+ Ad. Therefore, lim supt→∞N(t) ≤ A

das t → ∞. Since N(t) = Y (t) +X+(t) +224

X−(t), we can conclude that for t sufficiently large, 0 ≤ Y (t),X+(t) ≤ Ad.225

Similarly, from the fourth equation of the system (4.6) we have226

M (t) = µY (t− τ2) − µ0M(t).

This implies that M (t) + µ0M(t) = µY (t− τ2).

So M (t) + µ0M(t) ≤ µA

d, for t ≥ t0, for some t0 > 0.

Hence M(t) ≤ M(t0)e−µ(t−t0) +

µA

µ0d, for t ≥ t0,

so limsupt→∞M(t) ≤µA

µ0d.

It is straightforward to show that if P (t) is part of a solution of (4.6) then 0 ≤ P (t) ≤ 1. Hence,227

(Y (t),X+(t), N(t),M(t), P (t)) is ultimately bounded above.228

4.2.3. Equilibrium Analysis229

Now the equilibrium points (Y ∗,X∗+, N

∗,M∗, P ∗) of the delay model (4.6) satisfy230

18

β(N∗ −X∗+ − Y ∗)Y ∗ + β

1+β1M∗X∗+Y

∗ − (γ + α+ d)Y ∗ = 0,

λ(N∗ −X∗+ − Y ∗) M∗

k+M∗ + pγY ∗ − dX∗+ − λ0X

∗+P

∗ − β1+β1M∗X∗

+Y∗ = 0,

A− dN∗ − αY ∗ = 0,

µY ∗ − µ0M∗ = 0.

(4.7)

Here P ∗ will depend on τ1(≥ 0) through the following equation231

P ∗ (≡ F1, say) = e−

[

dτ1+λ0P ∗τ1+βY ∗τ1

1+β1M∗

]

(

≡ F2(P∗, τ1), say

)

. (4.8)

The expression on the righthand side (i.e. F2(P∗, τ1) ) is a decreasing function of τ1 such that F2(P

∗, 0) =232

1, F2(P∗,∞) = 0. Note that Y ∗ and M∗ depend on τ1 only through P ∗(τ1). So there exists at least233

one positive root (depending on τ1) of the transcendental equation (4.8) as P ∗ lies between 0 and 1. A234

graphical analysis to visualize this scenario is presented in Appendix B.235

4.3. Stability analysis and local Hopf bifurcation236

Case (a) : τ1 = τ2 = 0237

In absence of both delays the system (4.6) reduces to the system (3.2).238

Case (b) : τ1 = 0, τ2 > 0239

Then the system (4.6) reduces to the following system:240

dY

dt= β(N(t) −X+(t) − Y (t))Y (t) +

β

1 + β1M(t)X+(t)Y (t) − (γ + α+ d)Y (t),

dX+

dt= λ(N(t) −X+(t) − Y (t))

M(t)

k +M(t)+ pγY (t) − dX+(t) − λ0X+(t)

−β

1 + β1M(t)X+(t)Y (t),

dN

dt= A− dN(t) − αY (t),

dM

dt= µY (t− τ2) − µ0M(t).

(4.9)

19

It has the equilibrium point E∗(Y ∗,X∗+, N

∗,M∗) the same as the system (3.2). The variational matrix

at the endemic equilibrium E∗(Y ∗,X∗+, N

∗,M∗) is

J =

−M1 − ξ M2 M3 −M4

M5 −M6 − λ0 − ξ M7 M8

−M9 0 −M10 − ξ 0

µe−ξτ2 0 0 −M11 − ξ

.

Here M1 = βY ∗, M2 = −βY ∗ + βY ∗

1+β1M∗ , M3 = βY ∗, M4 =ββ1X∗

+Y ∗

(1+β1M∗)2 , M5 = − λM∗

k+M∗ + pγ −βX∗

+

1+β1M∗ ,241

M6 = λM∗

k+M∗ + d + λ0 + βY ∗

1+β1M∗ , M7 = λM∗

k+M∗ , M8 =λ(N∗−X∗

+−Y )k

(k+M∗)2+

ββ1X∗

+Y ∗

(1+β1M∗)2, M9 = α, M10 = d and242

M11 = µ0.243

The characteristic equation is244

ξ4 + (C1 +D1)ξ3 + (C2 +D2)ξ

2 + (C3 +D3)ξ + (C4 +D4)+

(E1ξ2 + (E2 + F1)ξ + (E3 + F2))e

−ξτ2 = 0.(4.10)

Here245

C1 = M1 +M6 +M10 +M11,246

C2 = −M2M5 +M1M6 +M6M10 +M1M10 +M3M9 +M6M11 +M1M11 +M10M11,247

C3 = −M2M5M10 +M1M6M10 +M3M6M9 +M2M7M9 −M2M5M11 +M1M6M11 +M6M10M11248

+M1M10M11 +M3M9M11,249

C4 = −M2M5M10M11 +M1M6M10M11 +M3M6M9M11 +M2M7M9M11,250

D1 = λ0,251

D2 = λ0(M10 +M11 +M1),252

D3 = λ0(M1M10 +M3M9 +M1M11 +M10M11),253

D4 = λ0(M3M9M11 +M1M10M11),254

E1 = µM4,255

E2 = −µ(−M4M10 +M2M8 −M4M6),256

E3 = −µ(M2M8M10 −M4M6M10),257

F1 = λ0µM4,258

F2 = λ0µM4M10.259

Theorem (4.1a) : The equilibrium point E∗ is locally asymptotically stable (LAS) for τ2 < τ20 where260

τ20 is the minimum positive value of261

20

τ20 = 1ω20

arccos

(E2+F1)ω220

[(C1+D1)ω220

−(C3+D3)]+(E1ω220

−E3−F2)[ω420

−(C2+D4)ω220

+(C4+D4)]

(E1ω220

−E3−F2)2+(E2+F1)2ω220

262

for ω20 corresponding to all positive real roots of (4.12). If the coefficients A1i (i = 1, 2, 3, 4) of equation263

(4.12) do not satisfy the Routh-Hurwitz conditions and (C4 +D4)2 < (E3 + F2)

2 holds then the delay τ2264

will not affect the stability of the system. If the coefficients A1i (i = 1, 2, 3, 4) of equation (4.12) satisfy265

the Routh-Hurwitz conditions then the system is LAS for all τ2 ≥ 0, provided that it is stable in the266

absence of delay.267

Proof : Put ξ = iω in (4.10) and separating real and imaginary parts we get268

(E1ω2 − E3 − F2) cos ωτ2 − (E2 + F1)ω sinωτ2 = ω4 − (C2 +D2)ω

2 + (C4 +D4),

(E1ω2 − E3 − F2) sinωτ2 + (E2 + F1)ω cosωτ2 = (C1 +D1)ω

3 − (C3 +D3)ω.(4.11)

Eliminating τ2 from (4.11) and put ω2 = ω1 we get269

ω41 +A11ω

31 +A12ω

21 +A13ω1 +A14 = 0, (4.12)

where270

A11 = (C1 +D1)2 − 2(C2 +D2),271

A12 = (C2 +D2)2 + 2(C4 +D4) − 2(C1 +D1)(C3 +D3) − E2

1 ,272

A13 = −2(C2 +D2)(C4 +D4) + (C3 +D3)2 + 2E1(E3 + F2) − (E2 + F1)

2,273

A14 = (C4 +D4)2 − (E3 + F2)

2.274

Case (b.1) : If the A1i (i = 1, 2, 3, 4) satisfy the Routh-Hurwitz conditions, then (4.12) has no positive275

real roots. In that case E∗ (if it exists) is LAS ∀τ2 > 0, provided that it is stable in the absence of delay,276

i.e. τ2 will not affect the stability of the system, when equation (4.12) has no positive real root.277

Case (b.2) : If the A1i (i = 1, 2, 3, 4) do not satisfy the Routh-Hurwitz conditions, in that case A14 < 0278

implies that equation (4.12) has at least one positive real root, i.e. if (C4 + D4)2 < (E3 + F2)

2 then279

equation (4.10) has a pair of purely imaginary roots say ±iω20 and for this value of ω20 we can get the280

value of τ2n from equation (4.11) as281

τ2n = 1ω20

arccos

(E2+F1)ω220

[(C1+D1)ω220

−(C3+D3)]+(E1ω220

−E3−F2)[ω420

−(C2+D4)ω220

+(C4+D4)]

(E1ω220

−E3−F2)2+(E2+F1)2ω220

+ 2nπω20

,

where n = 0, 1, 2, ... .282

By Butler’s lemma, [52] the endemic equilibrium remains stable for τ2 < τ20 . Without loss of generality283

suppose that ω20 represents the value of ω20 corresponding to τ20 .284

Theorem (4.1b) : If Φ1(ω20) > 0, the system (4.6) undergoes a Hopf Bifurcation at the positive equilib-285

rium as τ2 increases through τ20 , where the expression of Φ1(ω20) satisfies (4.13).286

21

Proof : Transversality condition for Hopf-bifurcation :287

Differentiating (4.10) with respect to τ2 we get288

289

dτ2dξ

= 4ξ3+3(C1+D1)ξ2+2(C2+D2)ξ+(C3+D3)E1ξ3+(E2+F1)ξ2+(E3+F2)ξ

eξτ2 + 2E1ξ+(E2+F1)E1ξ3+(E2+F1)ξ2+(E3+F2)ξ

− τ2ξ,290

291

Sgn[

d(Reξ)dτ2

]

τ2=τ20

= Sgn[

Re( dξdτ2

)−1]

ξ=iω20

,292

293

= Sgn

[

Re[−3(C1+D1)ω20

2+(C3+D3)] cos ω20τ2−[−4ω203+2(C2+D2)ω20 ] sin ω20τ2

−(E2+F1)ω202+iω20 [−E1ω20

2+(E3+F2)]+294

295

Re[−3(C1+D1)ω20

2+(C3+D3)] sinω20 τ2+[−4ω203+2(C2+D2)ω20 ] cos ω20τ2

−(E2+F1)ω202+iω20 [−E1ω20

2+(E3+F2)]i+296

297

Re2iω20E1+(E2+F1)

−(E2+F1)ω202+iω20 [−E1ω20

2+(E3+F2)]

]

,298

299

= Sgn[

−[−3(C1+D1)ω202+(C3+D3)]ω20 [(E2+F1)ω20 cos ω20τ2+(E1ω20

2−E3−F2) sinω20τ2]

(E2+F1)2ω204+ω20

2[−E1ω202+(E3+F2)]2

300

301

+[−4ω20

2+2(C2+D2)]ω202[(E2+F1)ω20 sinω20 τ2−(E1ω20

2−E3−F2) cos ω20τ2]

(E2+F1)2ω204+ω20

2[−E1ω202+(E3+F2)]2302

303

+ω20

2[−(E2+F1)2+2E1(−E1ω202+E3+F2)]

(E2+F1)2ω204+ω20

2[−E1ω202+(E3+F2)]2

]

.304

305

Using relation (4.11) we get the above expression as306

307

= Sgn[

[3(C1+D1)ω202−(C3+D3)][(C1+D1)ω20

2−(C3+D3)]+[4ω202−2(C2+D2)][ω20

4−(C2+D2)ω202+C4+D4]

(E2+F1)2ω202+[−E1ω20

2+(E3+F2)]2308

309

+−(E2+F1)2+2E1(−E1ω20

2+E3+F2)

(E2+F1)2ω202+[−E1ω20

2+(E3+F2)]2

]

,310

311

= Sgn[

4ω206+B1ω20

4+B2ω202+B3

(E2+F1)2ω202+[−E1ω20

2+(E3+F2)]2

]

,312

313

where314

B1 = 3(C1 +D1)2 − 6(C2 +D2),315

B2 = 2(C2 +D2)2 + 4(C4 +D4) − 4(C1 +D1)(C3 +D3) − 2E2

1 ,316

B3 = (C3 +D3)2 − 2(C2 +D2)(C4 +D4) − (E2 + F1)

2 + 2E1(E3 + F2).317

22

Let318

Φ1(ω20) = 4ω206 +B1ω20

4 +B2ω202 +B3. (4.13)

If Φ1(ω20) > 0 then Sgn[

d(Reξ)dτ2

]

τ2=τ20

> 0, i.e. the transversality condition holds and the system under-319

goes Hopf bifurcation.320

Case (c) : τ1 > 0, τ2 = 0321

The endemic equilibrium of the model (4.6) is E∗(Y ∗,X∗+, N

∗,M∗, P ∗) (see section 4.2.3). The variational

matrix at endemic equilibrium E∗(Y ∗,X∗+, N

∗,M∗, P ∗) is

J =

−M1 − ξ M2 M3 −M4 0

M5 −M6 −m1e−ξτ1 − ξ M7 M8 −M9

−M10 0 −M11 − ξ 0 0

m 0 0 −M12 − ξ 0

M13 −m2e−ξτ1 0 0 −M14 +m3e

−ξτ1 −M15 +m4e−ξτ1 − ξ

.

Here M1 = βY ∗, M2 = −βY ∗ + βY ∗

1+β1M∗ , M3 = βY ∗, M4 =ββ1X∗

+Y ∗

(1+β1M∗)2, M5 = − λM∗

k+M∗ + pγ −βX∗

+

1+β1M∗ ,322

M6 = λM∗

k+M∗ +d+ βY ∗

1+β1M∗ , M7 = λM∗

k+M∗ , M8 =λ(N∗−X∗

+−Y )k

(k+M∗)2 +ββ1X∗

+Y ∗

(1+β1M∗)2 , M9 = λ0X∗+, M10 = α, M11 = d,323

M12 = µ0, M13 = m2 = βP ∗

1+β1M∗ , M14 = m3 = ββ1Y ∗P ∗

(1+β1M∗)2, M15 = m1 = m4 = λ0P

∗ and m = µ.324

The characteristic equation is325

[ξ5 +A1ξ4 + (A2 + F1)ξ

3 + (A3 + F2)ξ2 + (A4 + F3)ξ + (A5 + F4)]e

ξτ1+

[C1ξ3 + C2ξ

2 + (B1 + C3)ξ + (B2 + C4)]e−ξτ1 + [D1ξ

4 +D2ξ3+

(D3 + E1)ξ2 + (D4 + E2)ξ + (D5 + E3)] = 0.

(4.14)

Here A1, A2, A3, A5, B1, B2, . . . F4 are given in Appendix A.326

Theorem (4.2a) : Let (A5 +B2 +C4 +F4)2 < (D5 +E3)

2 then the equilibrium E∗ is LAS for τ1 ∈ (0, τ10)327

where τ10 is the minimum positive value of328

τ10 = 1ω10

[

arccos(

−A22 A26+A23 A25

A21 A25+A22 A24

)

]

for ω10 corresponding to all positive real roots of (4.16) and the coefficients A2i (i = 1, 2, 3, 4, 5, 6) are329

described below, provided it is stable in the absence of delay.330

Proof : Put ξ = iω in (4.14) and separating real and imaginary parts we get331

A21 cosωτ1 −A22 sinωτ1 +A23 = 0,

A24 cosωτ1 +A25 sinωτ1 +A26 = 0,(4.15)

23

where332

A21 = A1ω4 − (A3 + C2 + F2)ω

2 + (A5 +B2 + C4 + F4),333

A22 = ω5 − (A2 − C1 + F1)ω3 + (A4 −B1 − C3 + F3)ω,334

A23 = D1ω4 − (D3 + E1)ω

2 + (D5 + E3),335

A24 = ω5 − (A2 + C1 + F1)ω3 + (A4 +B1 + C3 + F3)ω,336

A25 = A1ω4 − (A3 − C2 + F2)ω

2 + (A5 −B2 − C4 + F4),337

A26 = −D2ω3 + (D4 + E2)ω.338

Eliminating τ1 from (4.15) we get339

H1(ω) = (A21A25 +A22A24)2 − (A22A26 +A23A25)

2 − (A23A24 −A21A26)2 = 0. (4.16)

If (A5 + B2 + C4 + F4)2 − (D5 + E3)

2 < 0 then H1(0) < 0 and H1(∞) = +∞. So equation (4.16) has at340

least one positive real root ω10 .341

When ω = ω10 , equations (4.15) can be written as342

A21 cosω10τ1 −A22 sinω10τ1 +A23 = 0,

A24 cosω10τ1 +A25 sinω10τ1 +A26 = 0.(4.17)

HereA21, A22, A23, A24, A25 andA26 are obtained by substituting ω = ω10 intoA21, A22, A23, A24, A25 andA26.343

Equations (4.18) are simplified to give344

τ′

1n= 1

ω10

[

arccos(

−A22 A26+A23 A25

A21 A25+A22 A24

)

]

+ 2nπω10

; n = 0, 1, 2, ... ,

here iω10 is a purely imaginary root of equation (4.14).345

If (A5 + B2 + C4 + F4)2 − (D5 + E3)

2 < 0 then the equilibrium E∗(Y ∗,X∗+, N

∗,M∗, P ∗) is LAS for346

τ1 < τ10 . Without loss of generality suppose that ω10 represents the value of ω10 corresponding to τ10.347

Theorem (4.2b) : If Φ2(ω10) > 0, where Φ2(ω10) satisfies (4.18) the system (4.6) undergoes a Hopf348

Bifurcation at the positive equilibrium as τ1 increases through τ10 .349

Proof : Transversality condition for Hopf-bifurcation :350

Differentiating (4.14) with respect to τ1 , we get dτ1dξ

=351

[5ξ4+4A1ξ3+3(A2+F1)ξ2+2(A3+F2)ξ+(A4+F3)]eξτ1+[4D1ξ3+3D2ξ2+2(D3+E1)ξ+(D4+E2)]+[3C1ξ2+2C2ξ+(B1+C3)e−ξτ1 ]

[D1ξ5+D2ξ4+(D3+E1)ξ3+(D4+E2)ξ2+(D5+E3)ξ]+2[C1ξ4+C2ξ3+(B1+C3)ξ2+(B2+C4)ξ]e−ξτ1− τ1

ξ,352

353

Sgn[

d(Reξ)dτ1

]

τ1=τ10

= Sgn[

Re( dξdτ1

)−1]

ξ=iω10

= Sgn

[

Re P11+iP12G11+iG12

+Re iτ1ω10

]

= Sgn[

P11G11+P12G12

G211+G2

12

]

.354

355

24

P11, P12, G11 and G12 are given in Appendix A. Let356

Φ2(ω10) = P11G11 + P12G12. (4.18)

If Φ2(ω10) > 0 then Sgn[

d(Reξ)dτ1

]

τ1=τ10

> 0, i.e. the transversality condition holds and the system under-357

goes Hopf bifurcation.358

Case (d) : τ1 > 0 and τ2 fixed in (0, τ20)359

The endemic equilibrium of the model (4.6) is E∗(Y ∗,X∗+, N

∗,M∗, P ∗) (see section 4.2.3). The variational

matrix at the endemic equilibrium E∗(Y ∗,X∗+, N

∗,M∗, P ∗) is

J =

−M1 − ξ M2 M3 −M4 0

M5 −M6 −m1e−ξτ1 − ξ M7 M8 −M9

−M10 0 −M11 − ξ 0 0

me−ξτ2 0 0 −M12 − ξ 0

M13 −m2e−ξτ1 0 0 −M14 +m3e

−ξτ1 −M15 +m4e−ξτ1 − ξ

.

The characteristic equation is360

[ξ5 +A1ξ4 +A2ξ

3 +A3ξ2 +A4ξ +A5]e

ξτ1 + [B1ξ +B2]e−ξ(τ1+τ2)+

[C1ξ3 + C2ξ

2 + C3ξ + C4]e−ξτ1 + [D1ξ

4 +D2ξ3 +D3ξ

2 +D4ξ +D5]+

[E1ξ2 + E2ξ + E3]e

−ξτ2 + [F1ξ3 + F2ξ

2 + F3ξ + F4]eξ(τ1−τ2) = 0.

(4.19)

HereMi1 (i1 = 1−15), mi2 (i2 = 1−4),m, Ai3 (i3 = 1−5), Bi4 (i4 = 1−2), Ci5 (i3 = 1−4), Di6 (i6 = 1−5),361

Ei7 (i7 = 1 − 3), Fi8 (i8 = 1 − 4) are the same as described in Case (c).362

Theorem (4.3a) : Let (A5 + B2 + C4 + F4)2 < (D5 + E3)

2 and τ2 ∈ [0, τ20) then the equilibrium E∗ is363

LAS for τ1 ∈ (0, τ′

10) where364

τ′

10= 1

ω30

[

arccos(

−A32 A36+A33 A35

A31 A35+A32 A34

)

]

and the coefficients A3i (i = 1, 2, 3, 4, 5, 6) are described below.365

Proof : It is assumed that with equation (4.19), τ2 is in its stable interval and τ1 is considered as a366

parameter. Put ξ = iω in (4.19) and separating real and imaginary parts we get367

A31 cosωτ1 −A32 sinωτ1 +A33 = 0,

A34 cosωτ1 +A35 sinωτ1 +A36 = 0.(4.20)

Here368

A31 = [A1ω4−C2ω

3−A3ω2 +(A5 +C4)]+ [−F2ω

2 +(B2 +F4)] cos ωτ2 +[−F1ω3 +(B1 +F3)ω] sinωτ2,369

25

A32 = [ω5 − (A2 −C1)ω3 + (A4 − C3)ω] + [−F1ω

3 − (B1 − F3)ω] cos ωτ2 + [F2ω2 + (B2 − F4)] sinωτ2,370

A33 = [D1ω4 −D3ω

2 +D5] + [−E1ω2 + E3] cosωτ2 + E2ω sinωτ2,371

A34 = [ω5 − (A2 +C1)ω3 + (A4 + C3)ω] + [−F1ω

3 + (B1 + F3)ω] cos ωτ2 + [F2ω2 − (B2 + F4)] sinωτ2,372

A35 = [A1ω4 +C2ω

3−A3ω2 +(A5−C4)]+ [−F2ω

2− (B2−F4)] cos ωτ2 +[−F1ω3− (B1−F3)ω] sinωτ2,373

A36 = [−D2ω3 +D4ω] + E2ω cosωτ2 − [−E1ω

2 + E3] sinωτ2.374

Eliminating τ1 from (4.20) we get375

H2(ω) = (A31A35 +A32A34)2 − (A32A36 +A33A35)

2 − (A33A34 −A31A36)2 = 0. (4.21)

Note that if (A5 +B2 + C4 + F4)2 − (D5 + E3)

2 < 0 then H2(0) < 0 and H2(∞) = +∞.376

Now the above equation (4.21) is a transcendental equation in ω. The form of equation (4.21) is very377

complicated and it is difficult to predict the nature of its roots. Without going into detailed analysis with378

(4.21) it is assumed there exists at least one real positive root ω30 .379

When ω = ω30 , equation (4.20) can be written as380

A31 cosω30τ1 −A32 sinω30τ1 +A33 = 0,

A34 cosω30τ1 +A35 sinω30τ1 +A36 = 0,(4.22)

where A31, A32, A33, A34, A35, A36 are obtained by substituting ω = ω30 into A31, A32, A33, A34, A35 and381

A36.382

Equations (4.22) are simplified to give383

τ′

1n= 1

ω30

[

arccos(

−A32 A36+A33 A35

A31 A35+A32 A34

)

]

+ 2nπω30

; n = 0, 1, 2, ...

here iω30 is a purely imaginary root of equation (4.19).384

If (A5 +B2 +C4 +F4)2 < (D5 +E3)

2 and τ2 ∈ [0, τ20), then the equilibrium E∗(Y ∗,X∗+, N

∗,M∗, P ∗) is385

LAS for τ1 ∈ (0, τ′

10). Without loss of generality suppose that ω30 represents the value of ω30 corresponding386

to τ′

10.387

Theorem (4.3b) : If Φ3(ω30) > 0, the system (4.6) undergoes a Hopf Bifurcation at the positive equilib-388

rium as τ1 increases through τ′

10, where the expression of Φ3(ω30) satisfies (4.23).389

Proof : Transversality condition for Hopf-bifurcation :390

Differentiating (4.19) with respect to τ1 we get391

392

Sgn[

d(Reξ)dτ1

]

τ1=τ′

10

= Sgn[

Re( dξdτ1

)−1]

ξ=iω30

= Sgn

[

Re P21+iP22G21+iG22

+Reiτ

′

10ω30

]

= Sgn[

P21G21+P22G22

G221+G2

22

]

.393

394

26

Here P21, P22, G21 and G22 are given in the Appendix. Let395

Φ3(ω30) = P21G21 + P22G22. (4.23)

If Φ3(ω30) > 0 then Sgn[

d(Reξ)dτ1

]

τ1=τ′

10

> 0, i.e. the transversality condition holds and the system under-396

goes Hopf bifurcation.397

Case (e) : τ2 > 0 and τ1 fixed in (0, τ10)398

In a similar way as in Case (d) we can find the characteristic equation as399

[ξ5 +A1ξ4 +A2ξ

3 +A3ξ2 +A4ξ +A5] + [B1ξ +B2]e

−ξ(2τ1+τ2)+

[C1ξ3 + C2ξ

2 + C3ξ + C4]e−2ξτ1 + [D1ξ

4 +D2ξ3 +D3ξ

2 +D4ξ +D5]e−ξτ1+

[E1ξ2 + E2ξ + E3]e

−ξ(τ1+τ2) + [F1ξ3 + F2ξ

2 + F3ξ + F4]e−ξτ2 = 0.

(4.24)

Theorem (4.4a) : Let (A5 + C4 +D5)2 < (B2 + E3 + F4)

2 and τ1 ∈ [0, τ10) then the equilibrium E∗ is400

LAS for τ2 ∈ (0, τ′

20) where τ

′

20is the minimum value of401

τ′

20= 1

ω40

[

arccos(

−A42 A46+A43 A45

A41 A45+A42 A44

)

]

over ω40 corresponding to all positive real roots of (4.26) and the coefficients A4i, (i = 1, 2, 3, 4, 5, 6) are402

described below.403

Proof : It is considered that with equation (4.24), τ1 is in its stable interval and τ2 is considered as a404

parameter. Put ξ = iω in (4.24) and separating real and imaginary parts we get405

A41 cosωτ2 −A42 sinωτ2 +A43 = 0,

A44 cosωτ2 +A45 sinωτ2 +A46 = 0.(4.25)

Here406

A41 = [−F2ω2 + F4] − [E1ω

2 − E3] cosωτ1 +E2ω sinωτ1 +B2 cos 2ωτ1 +B1ω sin 2ωτ1,407

A42 = [F1ω3 − F3ω] −E2ω cosωτ1 − [E1ω

2 − E3] sinωτ1 −B1ω cos 2ωτ1 +B2 sin 2ωτ1,408

A43 = [A1ω4 −A3ω

2 +A5] + [D1ω4 −D3ω

2 +D5] cos ωτ1 − [D2ω3 −D4ω] sinωτ1409

−[C2ω2 − C4] cos 2ωτ1 − [C1ω

3 − C3ω] sin 2ωτ1,410

A44 = [−F1ω3 + F3ω] + E2ω cosωτ1 + [E1ω

2 − E3] sinωτ1 +B1ω cos 2ωτ1 −B2 sin 2ωτ1,411

A45 = [F2ω2 − F4] + [E1ω

2 − E3] cos ωτ1 − E2ω sinωτ1 −B2 cos 2ωτ1 −B1ω sin 2ωτ1,412

A46 = [ω5 −A2ω3 +A4ω] − [D2ω

3 −D4ω] cosωτ1 − [D1ω4 −D3ω

2 +D5] sinωτ1413

−[C1ω3 − C3ω] cos 2ωτ1 + [C2ω

2 − C4] sin 2ωτ1.414

Eliminating τ1 from (4.20) we get415

H2(ω) = (A42A46 +A43A45)2 + (A43A44 −A41A46)

2 − (A41A45 +A42A44)2 = 0. (4.26)

27

Note that if (A5 + C4 +D5)2 < (B2 + E3 + F4)

2 < 0 then H2(0) < 0 and H2(∞) = +∞.416

Again we assume that there exists at least one real positive root ω40. When ω = ω40 equation (4.25)417

can be written as418

A41 cosω40τ2 −A42 sinω40τ2 +A43 = 0,

A44 cosω40τ2 +A45 sinω40τ2 +A46 = 0,(4.27)

where A41, A42, . . . A46 are obtained by substituting ω = ω40 into A41, A42, . . . A46.419

Equations (4.27) are simplified to give420

τ′

2n= 1

ω40

[

arccos(

−A42 A46+A43 A45

A41 A45+A42 A44

)

]

+ 2nπω40

; n = 0, 1, 2, ...

here iω40 is a purely imaginary root of equation (4.24).421

If (A5 +C4 +D5)2 < (B2 +E3 +F4)

2 and τ1 ∈ [0, τ10), then the equilibrium E∗(Y ∗,X∗+, N

∗,M∗, P ∗) is422

LAS for τ2 ∈ (0, τ′

20). Without loss of generality suppose that ω40 represents the value of ω40 corresponding423

to τ′

20.424

Theorem (4.4b) : If Φ4(ω40) > 0, the system (4.6) undergoes a Hopf Bifurcation at the positive equilib-425

rium as τ2 increases through τ′

20, where Φ4(ω40) satisfies (4.28).426

Proof : Transversality condition for Hopf-bifurcation :427

Differentiating (4.24) with respect to τ2 we get428

429

Sgn[

d(Reξ)dτ2

]

τ2=τ′

20

= Sgn[

Re( dξdτ2

)−1]

ξ=iω40

= Sgn

[

Re P31+iP32G31+iG32

+Reiτ

′

20ω40

]

= Sgn[

P31G31+P32G32

G231+G2

32

]

,430

431

where P31, P32, G31 and G32 are given in the Appendix. Let432

Φ4(ω40) = P31G31 + P32G32. (4.28)

If Φ4(ω40) > 0 then Sgn[

d(Reξ)dτ2

]

τ2=τ′

20

> 0, i.e. the transversality condition holds and the system under-433

goes Hopf bifurcation.434

4.4. Permanence435

Biologically, persistence of a system means the survival of all populations of the system in future time.436

Mathematically, persistence of a system means that strictly positive solutions do not have omega limit437

points on the boundary of the non-negative cone. Butler, Freedman and Waltman [53], [54] developed438

the following definition of persistence:439

28

Definition 4.4.1. System (4.6) is said to be permanent if there are positive constants l, L such that each

positive solution (Y (t),X+(t), N(t),M(t), P (t)) of system (4.6) with initial conditions corresponding to

(4.3) satisfies

l ≤ limt→+∞

inf Y (t) ≤ limt→+∞

supY (t) ≤ L,

l ≤ limt→+∞

infX+(t) ≤ limt→+∞

supX+(t) ≤ L,

l ≤ limt→+∞

inf N(t) ≤ limt→+∞

supN(t) ≤ L,

l ≤ limt→+∞

infM(t) ≤ limt→+∞

supM(t) ≤ L,

l ≤ limt→+∞

inf P (t) ≤ limt→+∞

supP (t) ≤ L.

In order to prove permanence of system (4.6), we present the theory of permanence of infinite dimen-440

sional systems from Theorem 4.1 of Hale and Waltman [55]. Let X be a complete metric space. Suppose441

that X0 ∈ X, X0 ∈ X, X0⋂

X0 = ∅. Assume that T (t) is a C0 semigroup on X satisfying442

T (t) : X0 → X0,

T (t) : X0 → X0.(4.29)

Let Tb(t) = T (t)|X0 and let Ab be the global attractor for Tb(t).443

Lemma 4.4.1 [55]. Suppose that T (t) satisfies (4.29) and we have the following444

(i) there is a t0 ≥ 0 such that T (t) is compact for t > t0;445

(ii) T (t) is point dissipative in X;446

(iii) Ab =⋃

x∈Ab

w(x) is isolated and has an acyclic covering L, where

L = L1, L2, . . . , Ln;

(iv) W s(Li)⋂

X0 = ∅ for i = 1, 2, . . . , n.447

Then X0 is a uniform repeller with respect to X0, i.e., there is an ǫ0 > 0 such that, for any x ∈ X0,448

limt→+∞

inf d(T (t)x,X0) ≥ ǫ, where d is the distance of T (t)x from X0.449

Theorem 4.4.1. If βǫ0(γ+α+d) + 1 < R0 <

βǫ0+pγ+d+λ0

(γ+α+d) + 1, then the system (4.6) is permanent.450

Proof : We begin by showing that the boundary planes of R5+ repel the positive solutions to system (4.2)

uniformly. Let us define C0 to be

(ψ1, ψ2, ψ3, ψ4) ∈ C([−τ, 0],R4+ × [0, 1]) : ψ1(θ1) 6= 0, ψ2(θ1) = 0, ψ3(θ1) = 0, ψ4(θ1) = 0 and ψ5(θ1) = 0.

29

If C0 =intC([−τ, 0],R4+ × [0, 1]), it suffices to show that there exists an ǫ0 such that for all solutions ut of451

system (4.2) initiating from C0, limt→+∞

inf d(ut, C0) ≥ ǫ0. To this end we verify below that the conditions452

of Lemma 4.4.1 are satisfied. It is easy to see that C0 and C0 are positive invariant. Moreover, conditions453

(i) and (ii) of Lemma 4.4.1 are clearly satisfied. Thus, we only need to verify conditions (iii) and (iv).454

There is a constant solution E0 in C0. That is X−(t) = Ad, X+(t) = 0, Y (t) = M(t) = P (t) = 0. If455

(X−(t),X+(t), Y (t),M(t), P (t)) is a solution of system (4.2) initiating in C0, then X−(t) → Ad,X+(t) →456

0, Y (t) → 0,M(t) → 0 and P (t) → 0 as t→ ∞. It is obvious that E0 is isolated invariant.457

We now show that W s(E0)⋂

C0 = ∅. Assuming the contrary, i.e. W s(E0)⋂

C0 6= ∅, then there exists458

a positive solution (X−(t),X+(t), Y (t),M(t), P (t)) of the system (4.2) such that (Y (t),X+(t), N(t),M(t),459

P (t)) → (0, 0, Ad, 0, 0) as t → +∞. Let us choose ǫ0 > 0 small enough such that R0 > 1 + ǫ0. Let t0 > 0460

be sufficiently large such that Ad− ǫ0 < X−(t) < A

d+ ǫ0 for t > t0 − τ . Then we have, for t > t0,461

dYdt

≥ β(

Ad− ǫ0 −X+(t) − Y (t)

)

Y (t) + β1+β1M(t)X+(t)Y (t) − (γ + α+ d)Y (t). (4.30)

462

Hence dYdt

≥ β(

Ad− ǫ0 −X+(t) − Y (t)

)

Y (t) − (γ + α+ d)Y (t), (4.31)

463

or 1Y

dYdt

≥ β[(

Ad− ǫ0

)

−X+(t) − Y (t)]

Y (t) − (γ + α+ d). (4.32)

For X+, Y sufficiently small and R0 > 1 + βǫ0(γ+α+d) ,

1Y

dYdt

≥ ǫ1 > 0 for some ǫ1 > 0. Hence ∃t1 ≥ t0464

such that 1Y

dYdt

≥ ǫ1 > 0 for T ≥ t1. So Y (t) ≥ Y (t1)eǫ1(t−t1) for t ≥ t1 and Y (t1) > 0. This contradicts465

Y (t) → 0 as t → ∞. Therefore (Y (t),X+(t), N(t),M(t), P (t)) 9 (0, 0, Ad, 0, 0), which is a contradiction.466

Hence W s(E0)⋂

C0 = ∅. At this time, we are able to conclude from Lemma 4.4.1 that C0 repels the467

positive solutions of the system (4.2) uniformly, then the conclusion of Theorem 4.4.1 follows.468

5. Numerical simulations469

To observe the dynamics of the system, numerical experiments were carried out using Matlab. We470

base our parameters on the spread of Pneumococcus amongst children under two in Scotland [56]. Pneu-471

mococcus is a bacterial disease which has no permanent immunity. Hence an SIS model is suitable. We472

try to illustrate the analytical results of this paper with realistic parameter values although the objective473

is more to illustrate the analytical results rather than obtain accurate predictions.474

Lamb et al. estimate the size of the population at risk as N = 150, 000 and the per capita death rate475

as d = 1.3736 × 10−3 day−1 giving A = dN = 206.04 day−1. The infectious period 1γ

is given by Weir476

[57] as 1γ

= 7.1 weeks giving γ = 0.1408 week−1 = 0.02011 day−1. There is extremely low disease-related477

30

mortality from Pneumococcus carriage so we take α = 0.0 day−1. A Pneumococcus study by Zhang478

et al. [58] gives the basic reproduction number R0 to be in the range 1.8-2.2. We take R0 = 2 which479

then implies that β = 2.865 × 10−7 day−1. The remaining parameters are concerned with the disease480

awareness program and as we do not have the data on this these are estimated hypothetically as follows:481

λ = 0.9 day−1, λ0 = 0.3 day−1, µ = 1.3736 × 10−3 day−1, µ0 = 0.01 day−1, k = 500, β1 = 1 and p = 0.6.482

For the above set of parameter values we obtain E∗ = (1787.4, 73524, 150000, 245.51), σ1 = 0.6097 > 0,483

σ2 = 0.0068 > 0, σ3 = 3.1364 × 10−5 > 0, σ4 = 3.1425 × 10−8 > 0, σ1σ2 − σ3 = 0.0041 > 0 and484

σ1σ2σ3 − σ23 − σ2

1σ4 = 1.179 × 10−7 > 0. Hence this clearly indicates that for the above set of parameter485

values the system is LAS around the positive interior equilibrium. Figure 1 illustrates that, as expected,486

simulations carried out for a long time appear to converge to this equilibrium. For the above parameter487

values and initial conditions we observe that the solutions converge to the steady state in approximately488

three years. We repeated the simulations with the same parameters and other starting values and found489

similar behaviour and convergence times. Note that including environmental or demographic stochasticity,490

and seasonal forcing (or more than one of these together) might change the behaviour of the system.491

Next, we find the values of ∂Y ∗/∂µ, Y ∗ and ∂Y ∗/∂β1, Y∗ and plot them with respect to µ, β1 in492

Figure 2, 3 respectively. It is clear from Figure 2 and Figure 3 that if we increase either µ or β1 or both,493

the equilibrium number of infected individuals decreases, which confirms the result given in Remark 1.494

To study the impact of delays in system (4.2) we first fix τ1 = 0 days, and increase the value of τ2495

gradually. We observed that the system is LAS below a critical value τ20 (≈ 146 days, see Theorem 4.1) of496

τ2 and undergoes Hopf bifurcation as τ2 increases through τ20 (see Figure 4). For τ2 ≤ τ20 there is a unique497

LAS endemic equilibrium whose components are plotted on the y-axes in Figure 4. For τ > τ20 a stable498

limit cycle arises by Hopf bifurcation from this endemic equilibrium and Figure 4 plots the minimum and499

maximum values of these long-term stable limit cycle oscillations. Then we fixed τ1 = 120 days and drew500

the bifurcation diagram of the system (4.2) with respect to τ2, we observe that the system enters into501

limit cycle oscillation from a stable equilibrium as we increase the value of τ2 (see Figure 5). The system502

undergoes a Hopf bifurcation at τ2 ≈ 90 days (i.e. τ ′20≈ 90 days, see Theorem 4.4). Similarly, the system503

(4.2) loses its stability and enters into limit cycle oscillations through Hopf bifurcation at τ10 ≈ 128.4 days,504

when the second delay is absent (τ2 ≈ 0). In a similar way, keeping τ2 fixed at 60 days we observe that505

the system (4.2) undergoes a Hopf bifurcation at τ1 ≈ 134.7 days (i.e. τ ′10≈ 134.7 days, see Theorem506

4.3). In Figure 6 we have drawn the domain of the stability region with respect to τ1 and τ2 to visualize507

the impact of delays in the stability of the system (4.2).508

It is worth mentioning here that the interior equilibrium point of the system (4.6) depends on τ1,509

31

which is very different from traditional delay models. In traditional delay models the equilibrium points510

of the delay model and the non-delay model are the same. However in the present investigation, we have511

considered the survival probability (P ) in the interval of the time lag τ1 corresponding to aware people512

forgetting the impact of disease after this time lag. The equilibrium value of P depends on τ1 explicitly513

(see Appendix B). Consequently, the value of τ1 directly influences equilibrium population numbers. In514

Figure 7 we have plotted the equilibrium number of infected individuals, Y ∗, and the value of the survival515

probability, P ∗, against τ1. We observe that as τ1 increases the equilibrium number of infected individuals516

decreases.517

Our numerical computation also shows that for τ1 = 0 days, P ∗ = 1 and Y ∗ = 1787.4 and for τ1 = 180518

days, P ∗ = 0.051 and Y ∗ = 81.9. Therefore, it is clear that if the susceptible individuals become aware519

and remain aware for a long time then the equilibrium number of infected individuals decreases. However,520

we have also observed that for τ1 > τ10 (τ10 ≈ 128.4 days), the system shows limit cycle oscillation, which521

poses a challenge for controlling the epidemic outbreak.522

6. Conclusion523

In this paper we have considered the effect of disease awareness programs on disease dynamics where524

the susceptible population is divided into two different classes, aware susceptible and unaware susceptible.525

The model was considered first without any time delay and then with two time delays. The first time526

delay was due to people forgetting the impact of the disease after a time lag τ1. The second time delay was527

due to the media mounting a disease awareness campaign because of cases that had previously occurred528

after a time lag τ2.529

A differential equation model was used to examine the disease spread firstly with no time delay and530

then with a time delay. For the model with no time delay an expression for the basic reproduction number531

R0 was calculated. The DFE is LAS if and only if R0 < 1. For R0 > 1 the DFE becomes unstable and532

an endemic equilibrium exists.533

For the model with no time delay sufficient conditions for the endemic equilibrium to be LAS were534

derived. For the model with two time delays sufficient conditions for the stability of the endemic equi-535

librium and the existence of Hopf bifurcations were obtained for four different sets of values of the delay536

parameters, i.e. when τ1 = 0, τ2 > 0; τ2 = 0, τ1 > 0 and the two cases when τ1 > 0 and τ2 > 0 (see537

Theorems 4.1, 4.2, 4.3 and 4.4).538

Numerical simulations were performed to investigate the behavior of the system. They indicated that539

the system was LAS with realistic parameter values. We used the numerical simulations to visualize the540

32

effect of increasing time delays on the dynamics of the system.541

We observed that in our model if we increase the number of campaigns due to the awareness program542

then the disease transmission rate amongst the susceptible population declines. The numerical simulations543

also indicate that if the implementation rate of the awareness program increases then the equilibrium544

number of infected individuals decreases. We have also observed that if the time lag (τ1) in rejoining the545

unaware class of aware individuals increases, i.e. the susceptible individuals remain aware for a longer546

time, then the equilibrium number of infected individuals reduces. However, sustained oscillation may547

arise if the the time lag increases over a threshold value which could possibly pose a challenge in controlling548

the epidemic.549

However, the restrictions on the rate of immigration could have the ability to control the epidemic. It550

might be possible to control oscillations by controlling the rate of immigration [20]. Restricting immigra-551

tion might have a stabilizing effect on disease dynamics.552

In the present study we have considered the impact of an awareness campaign that acts on the553

whole population uniformly. This is a commonly made assumption in the literature on modeling media554

awareness campaigns. It would be appropriate for control of a disease that is established over a wide555

area. However it would not be appropriate for controlling a local outbreak of disease where an awareness556

campaign would have to be much more geographically focussed and act mostly on the local population. In557

those circumstances we would expect the impact of an awareness campaign to decrease as we move away558

from the epidemic outbreak or the number of infected individuals reduces. This would require a more559

sophisticated model and is a possible direction for future research. Note also that although the functional560

forms of the disease transmission term and the spread of information term have similarities we are not561

necessarily assuming the same transmission routes. Some other possible information transfer mechanisms562

could require fundamentally different information transmission terms [32]. This is also another potential563

direction for future work.564

Acknowledgement The authors are thankful to the anonymous reviewers and editor for their useful565

comments and suggestions. The research works of S. Samanta and T. Sardar are supported by Council of566

Scientific and Industrial Research (CSIR), Human Resource Development Group, New Delhi, India. The567

research of J. Chattopadhyay is supported by a DAE project (Ref No. 2/48(4)/2010-R&D II/8870).568

569

570

33

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38

0 500 1000 1500 20000

1000

2000

3000

Time

Infe

ctiv

e po

pula

tion

0 500 1000 1500 20002

4

6

8x 10

4

Time

Aw

ared

pop

ulat

ion

0 1000 20000

100

200

300

Time

Aw

aren

ess

prog

ram

Figure 1: Stable population distribution of (3.2) in absence of both delays (τ1 = τ2 = 0 days). Other parameter valuesare β = 2.8650 × 10−7 day−1, λ = 0.9 day−1, λ0 = 0.3 day−1, γ = 0.02011 day−1, d = 1.3736 × 10−3 day−1, µ = 1.3736 ×10−3 day−1, µ0 = 0.01 day−1, α = 0, k = 500, β1 = 1, A = 206.04 day−1, p = 0.6.

.

39

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−2000

−1500

−1000

−500

0

∂ Y

/∂ µ

µ

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

20

40

60

80

Y

µ

Figure 2: The figure depicts that the equilibrium number of infected individuals reduces with increasing µ (day−1) whereother parameter values are kept the same as in Figure 1.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−150

−100

−50

0

∂ Y

/∂ β

1

β1

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21760

1780

1800

1820

Y

β1

Figure 3: The figure depicts that the equilibrium number of infected individuals reduces with increasing β1.

40

0 100 2000

5000

10000

τ2

In

fec

tiv

e p

op

ula

tio

n

0 100 200

4

6

8

10

x 104

τ2

Aw

are

d p

op

ula

tio

n

0 100 2000

500

1000

τ2

Aw

are

ne

ss

pro

gra

m

Figure 4: Diagram showing Hopf bifurcation of system (4.2) with respect to τ2 (days) when τ1 = 0 days.

0 50 1000

500

1000

τ2

In

fec

tiv

e p

op

ula

tio

n

0 50 1000

5

10

15x 10

4

τ2

Aw

are

d p

op

ula

tio

n

0 50 1000

50

100

τ2

Aw

are

ne

ss

pro

gra

m

Figure 5: Diagram showing Hopf bifurcation of system (4.2) with respect to τ2 (days) when τ1 = 120 days.

41

0 20 40 60 80 100 120 140 160 180 0

20

40

60

80

100

120

140

160

180

τ1

τ 2

Figure 6: Domain of stability region with respect to τ1 (days) and τ2 (days) for the model (4.2). Other parameter values arekept the same as in Figure 1.

(a)0 20 40 60 80 100 120 140 160 180

0

200

400

600

800

1000

1200

τ1

Y*

(b)0 20 40 60 80 100 120 140 160 180

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

τ1

P*

Figure 7: Figures 7(a) and 7(b) show that the equilibrium number of infected individuals (Y ∗) and survival probability (P ∗)decrease for increase in τ1.

42

Appendix A704

Detailed mathematical expansions of terms in the paper.705

706

A.1 Terms in characteristic equation (4.14).707

A1 = M1 +M6 +M11 +M12 +M15,708

A2 = M1M11 +M1M12 +M1M15 +M11M12 +M11M15 +M12M15 +M3M10 +M1M6 −M2M5 +M6M11709

+M6M12 +M6M15,710

A3 = M1M11M12 +M1M11M15 +M1M12M15 +M11M12M15 +M3M10M12 +M3M10M15 +M1M6M11711

−M2M5M11 +M1M6M12 −M2M5M12 +M1M6M15 −M2M5M15 +M6M11M12 +M6M11M15712

+M6M12M15 +M2M7M10 +M3M6M10 +M2M9M13,713

A4 = M1M11M12M15+M3M10M12M15+M1M6M11M12−M2M5M11M12+M1M6M11M15−M2M5M11M15714

+M1M6M12M15−M2M5M12M15+M6M11M12M15+M2M7M10M12+M3M6M10M12+M2M7M10M15715

+M3M6M10M15 +M2M9M11M13 +M2M9M12M13,716

A5 = M1M6M11M12M15−M2M5M11M12M15+M2M7M10M12M15+M3M6M10M12M15+M2M9M11M12M13,717

B1 = −M4 m m1m4,718

B2 = −M11M4 m m1m4,719

C1 = −m1m4,720

C2 = −(M1 +M11 +M12)m1m4,721

C3 = −(M1M11 +M1M12 +M11M12 +M3M10)m1m4,722

C4 = −(M1M11M12 +M3M10M12)m1m4,723

D1 = m1 −m4,724

D2 = (M1 +M11 +M12 +M15)m1 − (M1 +M11 +M12 +M6)m4,725

D3 = (M1M11 +M1M12 +M1M15 +M11M12 +M11M15 +M12M15 +M3M10)m1726

−(M1M11 +M1M12 +M11M12 +M3M10 +M1M6 −M2M5 +M6M11 +M6M12)m4 −M2M9m2,727

D4 = (M1M11M12 +M1M11M15 +M1M12M15 +M11M12M15 +M3M10M12 +M3M10M15)m1728

−(M1M11M12 +M3M10M12 +M1M6M11 −M2M5M11 +M1M6M12 −M2M5M12 +M6M11M12729

+M2M7M10 +M3M6M10)m4 − (M2M9M11 +M2M9M12)m2,730

D5 = (M1M11M12M15 +M3M10M12M15)m1731

−(M1M6M11M12 −M2M5M11M12 +M2M7M10M12 +M3M6M10M12)m4 −M2M9M11M12m2,732

43

E1 = M4 m (m1 −m4),733

E2 = (M4M11m1 +M4M15m1 −M4M11m4 +M2M8m4 +M2M9m3 −M4M6m4) m,734

E3 = (M4M15m1 +M2M8m4 +M2M9m3 −M4M6m4)M11 m,735

F1 = M4 m,736

F2 = (M4M11 +M4M15 −M2M8 +M4M6) m,737

F3 = (M4M11M15 −M2M8M11 +M4M6M11 −M2M8M15 +M4M6M15 −M2M9M14) m,738

F4 = −(M2M8M15 −M4M6M15 +M2M9M14)M11 m.739

A.2 Terms in the transversality condition of Theorem 4.2b.740

P11 = [5ω410

− 3(A2 + F1 + C1)ω210

+ (A4 + F3 +B1 + C3)] cos ω10τ10741

+[4A1ω310

− 2(A2 + F2 − C2)ω10 ] sinω10τ10 + [−3D2ω210

+ (D4 + E2)],742

P12 = −[4A1ω103 − 2(A3 + F2 + C2)ω10 ] cosω10τ10743

+[5ω104 − 3(A2 + F1 − C1)ω

210

+ (A4 + F3 −B1 − C3)] sinω10τ10 + [−4D1ω310

+ 2(D3 + E1)ω10 ],744

G11 = 2[C1ω410

− (B1 + C3)ω210

] cosω10τ10 + 2[−C2ω310

+ (B2 + C4)ω10 ] sinω10τ10745

+[D2ω410

− (D4 + E2)ω210

],746

G12 = 2[−C2ω310

+ (B2 + C4)ω10 ] cosω10τ10 − 2[C1ω410

− (B1 + C3)ω210

] sinω10τ10747

+[D1ω510

− (D4 + E1)ω310

+ (D5 + E3)ω10].748

A.3 Terms in the transversality condition of Theorem 4.3b.749

P21 = [5ω430

− 3A2ω230

+A4] + [B1 − τ2B2] cos ω30(2τ′

10+ τ2) − τ2B1ω30 sinω30(2τ

′

10+ τ2)750

+[−3C1ω230

+C3] cos 2ω30τ′

10+2C2 sin 2ω30τ

′

10+[−3D2ω

230

+D4] cosω30τ′

10+[−4D1ω

330

+2D3ω30 ] sinω30τ′

10751

+[E2 − τ2(−E1ω230

+ E3)] cos ω30(τ′

10+ τ2) + [2E1ω30 − τ2E2ω30 ] sinω30(τ

′

10+ τ2)752

+[(−3F1ω230

+ F3) − τ2(−F2ω230

+ F4)] cosω30τ2 + 2[F2ω30 − τ2(−F1ω330

+ F3ω30)] sinω30τ2,753

P22 = [−4A1ω330

+ 2A3ω30] − τ2B1ω30 cosω30(2τ′

10+ τ2) − [B1 − τ2B2] sinω30(2τ

′

10+ τ2) + 2C2 cos 2ω30τ

′

10754

+[3C1ω230

− C3] sin 2ω30τ′

10+ [−4D1ω

330

+ 2D3ω30] cos ω30τ′

10+ [3D2ω

230

−D4] sinω30τ′

10755

+[2E1ω30 − τ2E2ω30 ]cosω30(τ′

10+ τ2) − [E2 + τ2(E1ω

230

− E3)] sinω30(τ′

10+ τ2)756

+[2F2ω30 + τ2(F1ω330

− F3ω30)] cos ω30τ2 + [(3F1ω230

− F3) + τ2(−F2ω230

+ F4)] sinω30τ2,757

G21 = −2B1ω230

cosω30(2τ′

10+ τ2) + 2B2ω30 sinω30(2τ

′

10+ τ2) + 2[C1ω

430

− C3ω230

] cos 2ω30τ′

10758

−2ω30 [C2ω230

− C4] sin 2ω30τ′

10+ ω2

30[D2ω

230

−D4] cosω30τ′

10+ ω30 [D1ω

430

−D3ω230

+D5] sinω30τ′

10759

44

−E2ω230

cosω30(τ′

10+ τ2) − ω30 [E1ω

230

−E3] sinω30(τ′

10+ τ2),760

G22 = 2B2ω30 cosω30(2τ′

10+ τ2) + 2B1ω

230

sinω30(2τ′

10+ τ2) − 2ω30 [C2ω

230

− C4] cos 2ω30τ′

10761

−2ω230

[C1ω230

− C3] sin 2ω30τ′

10+ ω30[D1ω

430

−D3ω230

+D5] cosω30τ′

10− ω2

30[D2ω

230

−D4] sinω30τ′

10762

−ω30[E1ω230

− E3] cos ω30(τ′

10+ τ2) + E2ω

230

sinω30(τ′

10+ τ2).763

A.4 Terms in the transversality condition of Theorem 4.4b.764

P31 = [5ω440

− 3A2ω240

+A4] + [B1 − 2τ1B2] cosω40(2τ1 + τ′

20) − 2τ1B1ω40 sinω40(2τ1 + τ

′

20)765

+[−3C1ω240

+ C3 + 2τ1(C2ω240

− C4)] cos 2ω40τ1 + [2C2ω40 + 2τ1(C1ω340

− C3ω40)] sin 2ω40τ1766

+[−3D2ω240

+D4−τ1(D1ω440−D3ω

240

+D5)] cos ω40τ1+[−4D1ω340

+2D3ω40+τ1(D2ω340−D4ω40)] sinω40τ1767

+[E2+τ1(E1ω240−E3)] cos ω40(τ1 + τ

′

20)+[2E1ω40−τ1E2ω40 ] sinω40(τ1 + τ

′

20)−[3F1ω

240−F3] cos ω40τ

′

20768

+2F2ω40 sinω40τ′

20,769

P32 = [−4A1ω340

+ 2A3ω40] − 2τ1B1ω40 cosω40(2τ1 + τ′

20) − [B1 − 2τ1B2] sinω40(2τ1 + τ

′

20)770

+[2C2ω40 + 2τ1(C1ω340

− C3ω40)] cos 2ω40τ1 + [3C1ω240

− C3 − 2τ1(C2ω240

− C4)] sin 2ω40τ1771

+[−4D1ω340

+2D3ω40+τ1(D2ω340−D4ω40)] cos ω40τ1+[3D2ω

240−D4+τ1(D1ω

440−D3ω

240

+D5)] sinω40τ1772

+[2E1ω40 − τ1E2ω40 ] cosω40(τ1 + τ′

20) − [E2 + τ1(E1ω

240

− E3)] sinω40(τ1 + τ′

20) + 2F2ω40 cosω40τ

′

20773

+[3F1ω240

− F3] sinω40τ′

20,774

G31 = −B1ω240

cosω40(2τ1 + τ′

20) +B2ω40 sinω40(2τ1 + τ

′

20) − E2ω

240

cosω40(τ1 + τ′

20)775

−[E1ω340

− E3ω40 ] sinω40(τ1 + τ′

20) + [F1ω

440

− F3ω240

] cosω40τ′

20− [F2ω

340

− F4ω40] sinω40τ′

20,776

G32 = B2ω40 cosω40(2τ1 + τ′

20) +B1ω

240

sinω40(2τ1 + τ′

20) − [E1ω

340

− E3ω40 ] cosω40(τ1 + τ′

20)777

+E2ω240

sinω40(τ1 + τ′

20) − [F2ω

340

− F4ω40] cos ω40τ′

20− [F1ω

440

− F3ω240

] sinω40τ′

20.778

45

Appendix B779

B.1 Numerical simulations to find the equilibria.780

781

In this appendix we obtain the equilibrium point of the equation (4.6) using the equations (4.7) and782

(4.8).783

First we fix parameters of the system (4.7) to be the same as in Figure 1 and vary P ∗ in the entire784

range within 0 and 1 to find (Y ∗,X∗+, N

∗,M∗) for each value of P ∗. Now we use equation (4.8) which is785

a transcendental equation in P ∗ to draw Figure 8. Let us consider the right hand side of the equation786

(4.8) as F2(P∗, τ1) = e

−

[

dτ1+λ0P ∗τ1+βY ∗τ1

1+β1M∗

]

. We fix τ1 and plot F2(P∗, τ1) for P ∗ lying between 0 and787

1. In Figure 8, we have taken some values of τ1 and plot F2, here blue, red, black and green solid curves788

correspond to the value of F2 at τ1 equal to 25, 50, 100 and 150 respectively. Lastly we plot the left hand789

side of the equation (4.8), i.e. F1(P∗) = P ∗ (the dashed blue line). The intersection between F1 and F2790

is the equilibrium value of P ∗ for different values of τ1.791

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

P*

F2

τ1=25

τ1=50

τ1=100

τ1=150

F2 = P

*

Figure 8: Graphical representation of equation (4.8) to find P ∗ for different τ1.

46